The examination of histological specimens obtained from core needle biopsies remains the definitive test for diagnosing prostate cancer (CaP). If cancer is present in these biopsy specimens, a surgeon may perform a radical prostatectomy (excise the prostate). Following prostatectomy, the prostate is sliced into whole-mount histological sections (WMHS). Staging and grading these WMHSs can help determine patient prognosis and treatment. Additionally, the spatial extent of CaP, as established by the analysis of WMHSs, can be registered to other modalities (eg. magnetic resonance imaging), providing a “ground truth” for the evaluation of computer-aided diagnosis (CAD) systems that operate on these modalities. The development of CAD algorithms for WMHSs is also significant for the following reasons: 1) CAD offers a viable means for analyzing the vast amount of the data present in WMHSs, a time-consuming task currently performed by pathologists, 2) the extraction of reproducible, quantifiable features can help refine our own understanding of prostate histopathology, thereby helping doctors improve performance and reduce variability in grading and detection, and 3) the data mining of quantified morphometric features may provide means for biomarker discovery, enabling, for example, the identification of patients whose CaP has a high likelihood of metastasis or post-treatment recurrence.
With respect to prostate histology, Begelman [49] presented a method for nuclei segmentation on hematoxylin and eosin (H&E) stained prostate tissue samples using a semi-supervised, fuzzy-logic engine. Adiga [50] presented a sophisticated system for the fast segmentation of cell nuclei in multispectral histological images. Doyle et al. used image texture features [51] and features derived from segmented nuclei and glands [52] to determine Gleason grade in core biopsy samples. Exploiting both domain-specific knowledge and low-level textural features, Naik et al. (References [53] and [54]) developed an automated system for segmenting glandular structures and discriminating between intermediate Gleason grades in core samples. To aid in manual cancer diagnosis, Gao [55] applied histogram thresholding to enhance the appearance of cytoplasm and nuclei. Hafiane [56] performed gland structure segmentation using a spatially constrained adaptation of fuzzy c-means and a multiphase level-set algorithm.
The most significant impediment in the development of automated systems for the analysis of MHSs is data volume. A typical whole-mount histological section digitized at 0.25 μm per pixel (equivalent to 40× magnification) contains approximately 200,000×60,000 pixels. This is 800 times the number pixels in a digitized mammogram. Consequently, nearly all previous systems (see References [84] to [95]) restrict their analysis to specific regions of interest (ROIs). These ROIs are typically less than one-thousandth the size of a WMHS. The only previous attempt—aside from our own—to identify CaP in relatively large HSs was presented by Diamond. Diamond divided a single WMHS into small 100×100 patches, classifying each patch individually using a single Haralick feature. However, the algorithm requires manual segmentation and classification of the glands. Additionally, execution time was 5.5 hours.
Contextual information can be invaluable in estimation tasks. For example, when determining a missing word in a document, clues can be ascertained from adjacent words, words in the same sentence, and even words in other sentences. In image processing tasks such as denoising, texture synthesis, and smoothing, estimating the intensity at any single pixel is facilitated by knowledge of the remaining pixel intensities. Unfortunately, modeling contextual information may be exceedingly difficult, especially if extensive dependencies exist among all sites (a generic term representing entities such as pixels or words). However, in many systems the preponderance of contextual information regarding an individual site is contained in those sites that neighbor it. These neighborhoods are often functions of spatial or temporal proximity. Since the contextual information in such systems can be restricted to local neighborhoods, modeling the systems becomes tractable.
In a Bayesian framework, the restriction of contextual information to local neighborhoods is called the Markov property, and a system of sites that obeys this property is termed a Markov random field (MRF). The merits of MRFs have been demonstrated for a variety of computer vision tasks such as clustering, and texture synthesis. MRFs have demonstrated particular proficiency in image segmentation and object detection tasks. Zhang [33], for example, used MRFs and expectation-maximization to segment magnetic resonance (MR) brain images. Farag [34] used a similar approach to segment multimodal brain and lung images. Li [35] applied MRFs to tumor detection in digital mammography. Employing an adaptive Markov model, Awate [36] presented an unsupervised method for classifying MR brain images. In general, MRF segmentation techniques have evolved into sophisticated algorithms that employ multiresolution analysis and complex boundary models.
MRFs are established through the construction of local conditional probability density functions (LCPDFs). These LCPDFs—one centered about each site—model the local inter-site dependencies of a random process. In combination, these LCPDFs can establish a joint probability density function (JPDF) relating all sites. However, only LCPDFs of certain functional forms will reconstitute a valid JPDF. Specifically, the Gibbs-Markov equivalence theorem states that the JPDF will be valid if (and only if) it, and transitively the LCPDFs, can be represented as Gibbs distributions. Unfortunately, constructing LCPDFs that simultaneously meet the following three conditions:
1) satisfy the Markov property;
2) combine to yield a valid JPDF; and
3) sufficiently model an underlying process
is nontrivial. Consequently, current MRF models are generic and/or heuristic, and thus, ignore crucial information. Currently, the computer vision literature advocates two disparate methods for constructing LCPDFs that meet these three conditions: parametric and nonparametric modeling.
Nonparametric modeling primarily focuses on condition (3). It is assumed, though the justification seems mostly empirical, that conditions (1) and (2) will be realized during the relaxation procedure (e.g. stochastic relaxation or iterated conditional modes). For example, in the case of texture reconstruction (i.e. constructing textural images from prescribed textural patterns), the LCPDFs—whose functional forms are initially unconstrained—are estimated from training images; the reconstruction algorithm attempts to rectify these unconstrained LCPDFs to satisfy conditions 1 and 2 during the actual synthesis (reconstruction). That is, the synthesizing procedure implicitly modifies the LDCPFs during texture generation, producing (hopefully) a random texture whose LCPDFs form a valid JPDF and whose appearance resembles the original texture. Regardless of the success of this process, the rectified LCPDFs are themselves not directly accessible (though they could perhaps be estimated by sampling the synthesized texture). Additionally, since we have no insight as to the possible functional forms of the LCPDFs, we can not anticipate whether or not the rectified LCDPFs will be able to describe our specific system. This “black box” type approach removes the insight typically provided by Bayesian modeling.
Parametric methods directly address conditions (1) and (2) by exploiting the Gibbs-Markov equivalence. That is, representing the JPDF as a Gibbs distribution guarantees that the attendant LCPDFs satisfy the Markov property and form a viable JPDF. However, satisfying condition 3 within the Gibbsian framework is difficult. Since Gibbs distributions are expressed as a product of potential functions, tailoring LCPDFs to model a specific process devolves into the intelligent selection of these functions. Unfortunately, potential functions are mathematical abstractions, lacking intuition. Consequently, constructing LCPDFs through their selection becomes an ad hoc procedure, usually resulting in generic and/or heuristic models.
Random fields, which are simply collections of random variables, provide a means for performing complex classification tasks within a Bayesian framework. Markov random fields—a particular type of random field—have proven useful in a variety of applications such as segmentation, texture synthesis. However, since random fields can assume a large number of states (classes), techniques for classifying them cannot rely on exhaustive searches and must employ more sophisticated techniques. Such techniques include simulated annealing, iterated conditional modes (ICM), loopy belief propagation, and graph cuts. However, these techniques, and nearly all other such methods, perform either maximum posterior marginals (MPM) or maximum a posteriori (MAP) estimation (or a variation of the two).
The weakness of MPM and MAP estimation lies in their inability to weight certain classification decisions more heavily than others—they implicitly weight all decisions equally. This ability is crucial for two reasons. First, some tasks naturally give rise to outcomes that are asymmetrical. For example, when identifying cancerous tissue, mislabeling a malignant lesion is far more egregious than misclassifying a benign structure. Second, many classification systems require the capability of adjusting their performance. For example, the performance of commercial systems for detecting mammographic abnormalities is typically adjusted to the highest detection sensitivity that incurs no more than two false positive per image. Unfortunately, since MRFs are restricted to MPM and MAP estimation, they are ill suited for such tasks.
Though others have addressed these problems [See References 1-10], they have used approaches that leverage low-level features (e.g. Haralick, wavelet, fractal). These features require extensive processing time, precluding the ability to analyze large images such as those found in histological analysis.